Geodesic
On the Bezdek–Pach Conjecture for Centrally Symmetric Convex Bodies
Canadian mathematical bulletin, Tome 52 (2009) no. 3, pp. 407-415

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DOI

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${{\mathbb{R}}^{d}}$ is ${{2}^{d}}$ . Naszódi proved that the quantity in question is not larger than ${{2}^{d+1}}$ . We present an improvement to this result by proving the upper bound $3\,\cdot \,{{2}^{d-1}}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.
DOI : 10.4153/CMB-2009-044-8
Mots-clés : 52C17, 51N20, 51K05, 52A21, 52A37, Bezdek–Pach conjecture, homothets, packing, Hadwiger number, antipodality 
Lángi, Zsolt; Naszódi, Márton. On the Bezdek–Pach Conjecture for Centrally Symmetric Convex Bodies. Canadian mathematical bulletin, Tome 52 (2009) no. 3, pp. 407-415. doi: 10.4153/CMB-2009-044-8
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     author = {L\'angi, Zsolt and Nasz\'odi, M\'arton},
     title = {On the {Bezdek{\textendash}Pach} {Conjecture} for {Centrally} {Symmetric} {Convex} {Bodies}},
     journal = {Canadian mathematical bulletin},
     pages = {407--415},
     year = {2009},
     volume = {52},
     number = {3},
     doi = {10.4153/CMB-2009-044-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-044-8/}
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